#145379

order	512
nilpotency class	4
isoclinism	isoclinic class of #144059
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1677
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

304 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(5,6,12,18)(7,19,25,55)(8,26,31,64)(9,16,34,52)(10,15,36,50)(11,37,33,17)(13,38,43,21)(14,44,48,29)(20,56,58,53)(22,41,24,54)(23,40,61,42)(27,65,63,67)(28,49,32,46)(30,47,35,51)(39,60,66,57)(45,59,68,62);(1,2)(3,4)(5,7)(6,13)(8,27)(9,23)(10,21)(11,22)(12,25)(14,45)(15,41)(16,19)(17,40)(18,43)(20,30)(24,33)(26,56)(28,59)(29,57)(31,63)(32,62)(34,61)(35,58)(36,38)(37,42)(39,47)(44,60)(46,67)(48,68)(49,65)(50,54)(51,66)(52,55)(53,64);(2,4)(5,8,9,32,33,44,36,30,12,31,34,28,11,29,10,35)(6,14,15,49,37,64,52,47,18,48,50,46,17,26,16,51)(7,20,38,57,22,62,23,63,25,58,21,60,24,59,61,27)(13,39,55,56,40,65,41,68,43,66,19,53,42,67,54,45)

Polycyclic presentation:

2^9|a^2=i,c^2=dh,d^2=gi,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,c^b=cf,d^b=dg,e^b=eg,f^b=fgi,g^b=gi,h^b=hi,d^c=di,e^c=egh,f^c=fi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52575);